## Total variation minimization approach in in-line x-ray phase-contrast tomography |

Optics Express, Vol. 21, Issue 10, pp. 12185-12196 (2013)

http://dx.doi.org/10.1364/OE.21.012185

Acrobat PDF (2738 KB)

### Abstract

The reconstruction problem in in-line X-ray Phase-Contrast Tomography is usually approached by solving two independent linearized sub-problems: phase retrieval and tomographic reconstruction. Both problems are often ill-posed and require the use of regularization techniques that lead to artifacts in the reconstructed image. We present a novel reconstruction approach that solves two coupled linear problems algebraically. Our approach is based on the assumption that the frequency space of the tomogram can be divided into bands that are accurately recovered and bands that are undefined by the observations. This results in an underdetermined linear system of equations. We investigate how this system can be solved using three different algebraic reconstruction algorithms based on Total Variation minimization. These algorithms are compared using both simulated and experimental data. Our results demonstrate that in many cases the proposed algebraic algorithms yield a significantly improved accuracy over the conventional L2-regularized closed-form solution. This work demonstrates that algebraic algorithms may become an important tool in applications where the acquisition time and the delivered radiation dose must be minimized.

© 2013 OSA

## 1. Introduction

1. R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys. **44**, 9 (2011) [CrossRef] .

*phase retrieval*problem, where the projected refraction index (i.e. linear phase and attenuation) of the specimen is retrieved independently for each recorded phase-contrast image and the problem of

*tomographic reconstruction*, where the three-dimensional distribution of the refraction index is computed from the collection of retrieved projections.

2. A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. **21**, 710–723 (2013) [CrossRef] .

## 2. Materials and methods

### 2.1. Single-distance phase retrieval

7. L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: demonstration of extended conditions with homogeneous objects,” Opt. Express **12**, 2960–2965 (2004) [CrossRef] [PubMed] .

*ϕ*(

*s*) is assumed to be proportional to the projected attenuation image

*μ*(

*s*). The proportionality ratio

*σ*=

*ϕ/μ*can be calculated as the ratio between the real and imaginary parts of the refractive index of the specimen. This allows us to express the Fourier transform of the phase-contrast image

*Ĩ*(

*w*) as the Fourier transform of the projected attenuation image

*μ̃*(

*w*) multiplied with the CTF that corresponds to the object-to-detector distance

*D*and the X-ray wavelength

*λ*: Here

*w*stands for the spatial frequency and

*δ*(

*w*) denotes the delta function at the origin of spatial frequency coordinates. A parallel monochromatic X-ray beam with a uniform illumination and intensity

*I*

_{0}= 1 was assumed for simplicity. This model is valid for chemically homogeneous or quasi-homogeneous objects (i.e.

*ϕ/μ*≈

*const*) with weak attenuation and slow-varying phase or for objects that are composed from light elements in a limited range of X-ray energies [8

8. D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. **206**, 33–40 (2002) [CrossRef] [PubMed] .

9. X. Wu and A. Yan, “Phase retrieval from one single phase-contrast x-ray image,” Opt. Express **17**, 11187 (2009) [CrossRef] [PubMed] .

*μ̃*(

*w*) from the observed phase-contrast image

*Ĩ*(

*w*) at frequencies

*w*that correspond to the zero-crossings of the CTF (see Fig. 1). Also, in the vicinity of each zero-crossing of the CTF the inverse problem based on Eq. (1) will be ill-posed due to measurement noise in addition to systematic linearization errors. At these frequencies

*μ̃*(

*w*) has to be computed using additional constraints that favor a particular solution based on

*a-priori*knowledge.

**and**

*μ***are defined in the spatial domain on a uniform grid of**

*I**k*values that belong to

*μ*(

*s*) and (

*I*(

*s*) − 1) respectively. A linear operator

*ℱ*will represent the uniform discrete Fourier transform. Element-wise multiplication with the CTF function is denoted by the linear operator

*𝒫*. Then Eq. (1) can be discretized in the following form: When the matrix notation is considered,

*𝒫*can be represented by a square

*k*×

*k*diagonal matrix: where

*P*(

*w*) = 2

*σ*sin(

*πλDw*

^{2}) − 2cos(

*πλDw*

^{2}). A more detailed explanation of the matrix notation is given in our paper on two-dimensional phase retrieval [2

2. A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. **21**, 710–723 (2013) [CrossRef] .

*P*(

*w*), we will exclude these terms from the system of equations. To do so we assume that the image

**is irrecoverable within the frequency bands |**

*μ**w*−

*w*

_{0}| <

*ε*, where

*w*

_{0}is the nearest zero-crossing of the CTF and

*ε*is a small constant that depends on the signal-to-noise ratio. Now an additional linear operator can be introduced: Operator

*𝒵*can be represented by a binary matrix and is applied to both sides of Eq. (2), making sure that terms corresponding to the small

*P*(

*w*) are set to zero: The system that we have obtained is underdetermined (it does not allow to determine (

*ℱ*

**) for |**

*μ**w*−

*w*

_{0}| <

*ε*) and does not have a unique solution unless additional constraints are added. Hofmann [10

10. R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express **19**, 25881–25890 (2011) [CrossRef] .

*ℱ*

**) = 0 for |**

*μ**w*−

*w*

_{0}| <

*ε*in order to solve the phase-retrieval problem. However, we expect that applying sparsity constraints to the joint phase-contrast-tomographic reconstruction will yield a better accuracy. A frequency dependent

*ε*can be defined when it is possible to estimate the power spectrum of the noise and the power spectrum of the reconstructed image modulated with the CTF

*a-priori*. Otherwise a constant factor

*ε*can be chosen, for instance, after evaluating the quality of the direct reconstruction proposed in [10

10. R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express **19**, 25881–25890 (2011) [CrossRef] .

### 2.2. Tomography

*s*to describe the transverse coordinate an observed image of the object and

*w*as its counterpart in the Fourier domain. A new coordinate

*θ*will be introduced for the angle at which the image is recorded during the tomographic acquisition. Coordinates (

*x*,

*y*) will be used to describe a 2D tomographic image of the object (see Fig. 1).

*f*(

*x*,

*y*) which is defined on ℝ

^{2}→ ℂ as a square integrable function with bounded support. A linear projection of

*f*(

*x*,

*y*) in the direction

*θ*will be defined along the coordinate

*s*as: According to the

*central slice theorem*, if a two-dimensional Fourier transform of the image

*f*(

*x*,

*y*) is computed values of

*f̃*(

*u*,

*v*) that lie on a radial line passing through the center of coordinates under an angle

*θ*(central slice) will correspond to the one-dimensional Fourier transform taken along the

*s*coordinate of the projection

*p*(

*θ*,

*s*): The central slice theorem demonstrates an important relation between the Radon transform and the Fourier transform of the two-dimensional image

*f*(

*x*,

*y*). It follows from Eq. (9) that the function

*f*(

*x*,

*y*) can be sampled in 2D Fourier space using 1D Fourier transforms of its own projections

*p*(

*θ*,

*s*). This facilitates direct reconstruction based on the inverse Fourier transform of

*f̃*(

*u*,

*v*) which has to be computed using interpolation. Moreover, it was shown in [4

4. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory **52**, 489–509 (2006) [CrossRef] .

*f̃*(

*u*,

*v*), i.e. using very few projections. The latter has important consequences for regularization of the phase-retrieval problem described in the following subsection.

### 2.3. Phase-contrast tomography

*Ĩ*(

*θ*,

*w*) is the one-dimensional Fourier transform of the phase-contrast sinogram. According to Eq. (10),

*f̃*(

*u*,

*v*) is undetermined for frequencies that correspond to zero-crossings of the CTF. And since the accuracy of the linear approximation and the observations

*Ĩ*(

*θ*,

*w*) is finite,

*f*̃(

*u*,

*v*) can only be computed outside of the circular bands that correspond to |

*w*−

*w*

_{0}|<

*ε*(Fig. 1). Thus, the described problem of the tomographic reconstruction based on single-distance phase-contrast data is underdetermined.

**is composed from**

*f**m*samples of

*f*(

*x*,

*y*) defined on a Cartesian grid. Projections

*p*(

*θ*,

*s*) are sampled on a regular grid of

*n*elements stored in a vector

**. Using the Radon transform operator**

*p**ℛ*we can write a discrete representation of Eq. (7): Here

*ℛ*can be represented by an

*n*×

*m*matrix. This system can be solved using either an approximation of the

*ℛ*

^{−1}(e.g. filtered back-projection) or in least-squares sense using one of the algebraic reconstruction algorithms (e.g. EM, ART, SIRT) [11].

*𝒵*,

*𝒫*and

*ℱ*are applied to vectors of

*n*elements that contain all projections generated by

*ℛ*

**instead of a single projection of**

*f**k*elements as it is in Eq. (6). Their matrix representations can be easily constructed by stacking the corresponding “single projection” matrices into a block diagonal matrix of

*n*×

*n*elements.

### 2.4. Preconditioning

*w*−

*w*

_{0}| >

*ε*, Eq. (1) can be accurately computed using direct inversion. Then Eq. (12) can be rewritten as follows: where

**is a vector representation of the projected attenuation of the object, which can be calculated from the phase-contrast sinogram**

*μ***using direct phase retrieval. Algebraic methods based on Eq. (13) are likely to converge faster than the ones based on Eq. (12), since the inversion of the**

*I**𝒫*operator is already solved during the phase retrieval step. However, unlike the approach, where the tomographic reconstruction is computed subsequently after phase retrieval, Eq. (13) allows us to take into account that the information about the attenuation of the object is lost at spatial frequencies |

*w*−

*w*

_{0}| <

*ε*. Such an approach can also be used when the number of tomographic projections is limited.

*ℛ*

^{−1}is the approximated discrete inverse Radon transform (e.g. filtered back-projection),

*ℱ̂*is the two-dimensional discrete Fourier transform and

*𝒫̂*represents an elementwise multiplication with the discrete version of the two-dimensional CTF = 2

*σ*sin(

*πλD*(

*u*

^{2}+

*v*

^{2})) −2cos(

*πλD*(

*u*

^{2}+

*v*

^{2})). Equation (14) permits application of algebraic algorithms with additional constraints to the phase retrieval problem while the problem of tomographic reconstruction is solved in a non-iterative manner. This approach can also be faster than calculation based on Eq. (12), since it does not require recalculation of the back and forward Radon transform for each iteration.

### 2.5. Algebraic methods

*𝒜*=

*𝒵𝒫ℱℛ*and

**=**

*Ĩ**𝒵ℱ*

**. Using gradient descent,**

*I***is computed according to the following iterative scheme: where**

*f**j*is the iteration number,

*𝒜*is the transpose of

^{T}*𝒜*, so

*𝒜*=

^{T}*ℛ*. The constant

^{T}ℱ^{T}𝒫^{T}𝒵^{T}*α*represents the step size in the opposite gradient direction. In order to guarantee convergence, the constant

*α*has to be sufficiently small and can either be calculated using an additional line-search step or from the eigenvalues of (

*𝒜*) [12

^{T}𝒜12. L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math. **16**, 1–3 (1966) [CrossRef] .

*ℱ*is orthogonal, so

*ℱ*=

^{T}*ℱ*

^{−1}; operators

*𝒵*and

*𝒫*are represented by diagonal matrices, so

*𝒵*=

^{T}*𝒵*,

*𝒫*=

^{T}*𝒫*, and

*ℛ*represents the unfiltered back-projection operator.

^{T}**can often be computed accurately from severely underdetermined tomographic system using methods with additional constraints, such as TV minimization. In the TV minimization approach an additional term is added to the objective function: where**

*f**and ∇*

_{h}*are the horizontal and vertical finite difference operators and*

_{v}*λ*denotes the weight of the regularization term. Depending on the magnitude of

_{TV}*λ*the solution will be promoted either towards greater conformity with the observed data or towards greater sparsity of the gradient magnitude. Equation (17) represents a non-smooth convex minimization problem which can be solved using one of the iterative TV minimization methods [13

_{TV}13. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision **20**, 89–97 (2004) [CrossRef] .

14. J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. **53**, 67–92 (2010) [CrossRef] .

15. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci. **2**, 183–202 (2009) [CrossRef] .

2. A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. **21**, 710–723 (2013) [CrossRef] .

*full algebraic reconstruction*, the ones based on Eq. (13) -

*algebraic tomographic reconstruction*and Eq. (14) -

*algebraic phase retrieval*.

## 3. Simulations

*f*(

*x*,

*y*) (Fig. 2(a)). The wave function of the object

*T*(

*θ*,

*s*) was computed for each tomographic angle

*θ*using the following expression: were

*δ*denotes the decrement of the complex refractive index,

*β*stands for the attenuation index and

*f*(

*x*,

*y*) denotes a dimensionless normalized attenuation of the digital phantom. Subsequently, the intensity images were generated using the following expression: where the Fresnel propagator is represented by

*P*=

_{λ}*e*

^{iπλDw2}and the Optical Transfer Function (OTF) of the imaging system is modeled by a Gaussian function

*a*(Fig. 2(c)). Figure 2(e) shows the result of direct application of filtered-back projection to the raw phase-contrast images. The result of the sequential reconstruction is shown on Fig. 2(d) and Fig. 2(f). The retrieved phase images (Fig. 2(d)) suffers from typical artifacts –low frequency noise and fringes around large variations of intensity. These artifacts correspond to spatial frequencies at which the CTF has low amplitude and the reconstruction errors are high. They are propagated into the subsequent tomographic reconstruction (Fig. 2(f)).

*λ*= 0.31

*Å*(energy 40 KeV), propagation length

*D*= 1 m and a pixel size of 1

*μ*m. Noise was added to the sinogram after modeling the complete imaging chain. Since the intensity is varying only moderately across the sinogram, an additive Gaussian noise term was assumed to be a good model. Parameters used in various simulations are shown in Table 1.

*sequential approach*(phase retrieval followed by the filtered back-projection),

*unconstrained algebraic reconstruction*based on Eq. (12) and three algorithms based on TV minimization:

*full algebraic reconstruction*- Eq. (12),

*algebraic tomographic reconstruction*- Eq. (13),

*algebraic phase retrieval*- Eq. (14). A known Gaussian OTF was included in all algebraic reconstruction models in order to achieve a sharper reconstruction. Reconstructions were performed on a 512 × 512 pixels grid in order to avoid boundary effects. The same stopping condition was used for all algebraic methods: The weight of the regularization term

*λ*had to be adjusted depending on the underlying linear system and the variance of the simulated data. Automated methods for determining the regularization parameters produce a wide spread of results depending on the method [16

_{TV}16. G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy **198**, 63–75 (2000) [CrossRef] .

*λ*varied in the range from 10

_{TV}^{−7}(for the “blur” simulation) to 10

^{−5}(for the “strong phase” and “realistic” simulations). For algebraic tomographic reconstruction

*λ*was in the range from 10

_{TV}^{−10}to 10

^{−8}. And for algebraic phase retrieval we used

*λ*ranging from 10

_{TV}^{−4}to 10

^{−2}. The constant parameter

*ε*was set to 10

^{4}

*m*

^{−1}for simulations with small errors (“weak phase”, “few projections” and “blur”) and 10

^{5}

*m*

^{−1}for simulations with large errors (“strong phase”, “noise” and “realistic”). Figure 3 illustrates the error magnitude associated to the resulting reconstructions. The corresponding Root Mean Square Error (RMSE) can be found in Table 2.

## 4. Experiments

*full algebraic reconstruction*and the

*algebraic phase retrieval*methods applied to an experimental X-ray PCT dataset. The data was collected at the beamline ID11 of the European Synchrotron Radiation Facility (Grenoble, France). The tomographic scan was acquired in-situ for spherical polycrystalline copper (Makin Metal Powders (UK) Ltd., diameter 50

*μ*m) during sintering at 1050°C. The specimen was placed in a quartz capillary with a 500

*μ*m internal diameter. During the experiment gas shielding (argon: 98% and hydrogen: 2%) was applied. The scan was performed in a continuous 180° mode with 650 projections. Phase-contrast images were acquired using an X-ray beam with a mean energy E = 40 KeV (Δ

*E/E*= 10

^{−3}), a source to object distance of 96 m and a propagation distance of 25 cm. The size of each image was 512 × 256 pixels with a pixel size of 1.4 × 1.4

*μm*

^{2}.

*μ*m. We varied the number of iterations depending on the rough estimate of the convergence speed of a particular algorithm. In full algebraic reconstruction we used 2000 iterations. In algebraic phase retrieval based on 650 tomographic projections 300 iterations were used. Two times more iterations were used in both approaches when applied to 65 tomographic projections.

## 5. Conclusion

6. X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging **2**, 455–484 (2008) [CrossRef] .

18. G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. **32**, 654–665 (2005) [CrossRef] [PubMed] .

**21**, 710–723 (2013) [CrossRef] .

19. P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. **75**, 2912–2914 (1999) [CrossRef] .

20. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. **32**, 1617–1619 (2007) [CrossRef] [PubMed] .

## Acknowledgments

## References and links

1. | R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys. |

2. | A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr. |

3. | F. Natterer, |

4. | E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory |

5. | M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc. |

6. | X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging |

7. | L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: demonstration of extended conditions with homogeneous objects,” Opt. Express |

8. | D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. |

9. | X. Wu and A. Yan, “Phase retrieval from one single phase-contrast x-ray image,” Opt. Express |

10. | R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express |

11. | A. C. Kak and M. Slaney, |

12. | L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math. |

13. | A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision |

14. | J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg. |

15. | A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci. |

16. | G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy |

17. | S.-R. Zhao and H. Halling, “A new Fourier method for fan beam reconstruction,” IEEE Nucl. Sci. Symp. Med. and Imaging Conf. |

18. | G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. |

19. | P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett. |

20. | J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(100.6950) Image processing : Tomographic image processing

(110.7440) Imaging systems : X-ray imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 11, 2013

Revised Manuscript: March 29, 2013

Manuscript Accepted: March 30, 2013

Published: May 10, 2013

**Virtual Issues**

Vol. 8, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Alexander Kostenko, K. Joost Batenburg, Andrew King, S. Erik Offerman, and Lucas J. van Vliet, "Total variation minimization approach in in-line x-ray phase-contrast tomography," Opt. Express **21**, 12185-12196 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12185

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### References

- R. C. Chen, L. Rigon, and R. Longo, “Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data,” J. Phys. D Appl. Phys.44, 9 (2011). [CrossRef]
- A. Kostenko, K.J. Batenburg, H. Suhonen, S.E. Offerman, and L.J. van Vliet, “Phase retrieval in in-line x-ray phase-contrast imaging based on total variation minimization,” Opt. Expr.21, 710–723 (2013). [CrossRef]
- F. Natterer, The Mathematics of Computerized Tomography( New York: Wiley, 1986).
- E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006). [CrossRef]
- M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Proc.50, 1417–1428 (2003). [CrossRef]
- X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inv. Probl. and Imaging2, 455–484 (2008). [CrossRef]
- L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: demonstration of extended conditions with homogeneous objects,” Opt. Express12, 2960–2965 (2004). [CrossRef] [PubMed]
- D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002). [CrossRef] [PubMed]
- X. Wu and A. Yan, “Phase retrieval from one single phase-contrast x-ray image,” Opt. Express17, 11187 (2009). [CrossRef] [PubMed]
- R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express19, 25881–25890 (2011). [CrossRef]
- A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (IEEE Press, 1988).
- L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Math.16, 1–3 (1966). [CrossRef]
- A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004). [CrossRef]
- J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010). [CrossRef]
- A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci.2, 183–202 (2009). [CrossRef]
- G. M. P. van Kemplen and L. J. van Vliet, “The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms,” J. of Microscopy198, 63–75 (2000). [CrossRef]
- S.-R. Zhao and H. Halling, “A new Fourier method for fan beam reconstruction,” IEEE Nucl. Sci. Symp. Med. and Imaging Conf.2, 1287–1291 (1995).
- G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys.32, 654–665 (2005). [CrossRef] [PubMed]
- P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999). [CrossRef]
- J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett.32, 1617–1619 (2007). [CrossRef] [PubMed]

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